# The New Similarity Measure and Distance Measure of a Hesitant Fuzzy Linguistic Term Set Based on a Linguistic Scale Function

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## Abstract

**:**

## 1. Introduction

- (1)
- In order to overcome the disadvantage of the similarity measure proposed by Liao et al. [31], a new similarity measure combining the existing cosine similarity measure [31] and the Euclidean distance measure of HFLTSs is proposed in this paper, which can improve the accuracy of the calculation to some extent.
- (2)
- On the basis of the linguistic scale function, the paper proposes a new similarity measure between two HFLTSs; it is already known that the linguistic scale function can improve the flexibility of the transformation of the linguistic decision information in different semantic environments. The proposed method is capable of expressing the fuzzy linguistic information more flexibly and improving the adaptability of HFLTSs in practice.
- (3)
- According to the relationship between the similarity measure and the distance measure, this paper proposes a new distance measure of HFLTSs and extends the TOPSIS method to it; it focuses on the differences between different alternatives, which can improve the effectiveness of solving MCDM problems.

## 2. Preliminaries

#### 2.1. MCDM

#### 2.2. LTS

**Definition**

**1.**

- (1)
- The LTS$S$satisfies the following properties:
- (2)
- The set$S$is ordered:${s}_{i}\le {s}_{j}$if$i\le j$;$max\left({s}_{i},{s}_{j}\right)={s}_{i}$if${s}_{i}\ge {s}_{j}$;$min\left({s}_{i},{s}_{j}\right)={s}_{i}$if${s}_{i}\le {s}_{j}$;
- (3)
- The negation operator is defined:$neg\left({s}_{i}\right)={s}_{j}$. satisfying with$i+j=2t$.

#### 2.3. HFLTS

**Definition**

**2.**

**Example**

**1.**

**Definition**

**3.**

**Lemma**

**1.**

- (1)
- ${H}_{S}^{1}>{H}_{S}^{2}$if and only if$F\left({H}_{S}^{1}\right)>F\left({H}_{S}^{2}\right)$;
- (2)
- ${H}_{S}^{1}={H}_{S}^{2}$if and only if$F\left({H}_{S}^{1}\right)=F\left({H}_{S}^{2}\right)$.

#### 2.4. Existing Distance and Similarity Measures Between HFLTSs

**Definition**

**4.**

**Remark**

**1.**

**Definition**

**5.**

**Remark**

**2.**

#### 2.5. Linguistic Scale Function

**Definition**

**6.**

**Example**

**2.**

**Remark**

**3.**

## 3. The Score Function, Similarity Measure, and Distance Measure Between HFLTSs Based on a Linguistic Scale Function

#### 3.1. The Score Function Between HFLTSs Based on the Linguistic Scale Function

**Definition**

**7.**

**Theorem**

**1.**

- (1)
- If${F}^{\ast}\left({H}_{S}^{1}\right)>{F}^{\ast}\left({H}_{S}^{2}\right)$, then${H}_{S}^{1}>{H}_{S}^{2}$;
- (2)
- If${F}^{\ast}\left({H}_{S}^{1}\right)={F}^{\ast}\left({H}_{S}^{2}\right)$, then${H}_{S}^{1}={H}_{S}^{2}$.

**Example**

**3.**

#### 3.2. The Similarity Measure Between HFLTSs Based on the Linguistic Scale Function

**Lemma**

**2.**

- (1)
- $0\le S\left({H}_{S}^{1},{H}_{S}^{2}\right)\le 1$,
- (2)
- $S\left({H}_{S}^{1},{H}_{S}^{2}\right)=1$if and only if${H}_{S}^{1}={H}_{S}^{2}$,
- (3)
- $S\left({H}_{S}^{1},{H}_{S}^{2}\right)=S\left({H}_{S}^{2},{H}_{S}^{1}\right)$.

**Example**

**4.**

**Definition**

**8.**

**Theorem**

**2.**

- (1)
- $0\le {D}_{\omega HFL}^{\prime}\left({H}_{S}^{1},{H}_{S}^{2}\right)\le 1$;
- (2)
- ${D}_{\omega HFL}^{\prime}\left({H}_{S}^{1},{H}_{S}^{2}\right)=0$if and only if${H}_{S}^{1}={H}_{S}^{2}$;
- (3)
- ${D}_{\omega HFL}^{\prime}\left({H}_{S}^{1},{H}_{S}^{2}\right)={D}_{\omega HFL}^{\prime}\left({H}_{S}^{2},{H}_{S}^{1}\right)$.

**Proof.**

**Remark**

**4.**

**Definition**

**9.**

**Remark**

**5.**

**Definition**

**10.**

**Theorem**

**3.**

**Proof.**

- (1)
- Since $0\le f\le 1,Co{s}_{HFL}^{\prime}\left({H}_{S}^{1},{H}_{S}^{2}\right)$ can be considered as the extension of cosine function, then $0\le Co{s}_{HFL}^{\prime}\left({H}_{S}^{1},{H}_{S}^{2}\right)\le 1$. According to Theorem 2, we know that ${D}_{HFL}^{\prime}\left({H}_{S}^{1},{H}_{S}^{2}\right)$ is a distance measure, then $0\le 1-{D}_{HFL}^{\prime}\left({H}_{S}^{1},{H}_{S}^{2}\right)\le 1$. Thus, we get $0\le Co{s}_{HFL}^{\prime}\left({H}_{S}^{1},{H}_{S}^{2}\right)+1-{D}_{HFL}^{\prime}\left({H}_{S}^{1},{H}_{S}^{2}\right)\le 2$, so $0\le {S}_{HFL}^{\ast}\left({H}_{S}^{1},{H}_{S}^{2}\right)\le 1$ is obvious.
- (2)
- If ${H}_{S}^{1}={H}_{S}^{2}$, we have $f({s}_{{\delta}_{l}^{1}}\left({x}_{j}\right))=f({s}_{{\delta}_{l}^{2}}\left({x}_{j}\right))$, $Co{s}_{HFL}^{\prime}\left({H}_{S}^{1},{H}_{S}^{2}\right)=1$, ${D}_{HFL}^{\prime}\left({H}_{S}^{1},{H}_{S}^{2}\right)=0$, then ${S}_{HFL}^{\ast}\left({H}_{S}^{1},{H}_{S}^{2}\right)=1$. On the other hand, when ${S}_{HFL}^{\ast}\left({H}_{S}^{1},{H}_{S}^{2}\right)=1$, we have $Co{s}_{HFL}^{\prime}\left({H}_{S}^{1},{H}_{S}^{2}\right)+1-{D}_{HFL}^{\prime}\left({H}_{S}^{1},{H}_{S}^{2}\right)=2$; that is, $Co{s}_{HFL}^{\prime}\left({H}_{S}^{1},{H}_{S}^{2}\right)=1+{D}_{HFL}^{\prime}\left({H}_{S}^{1},{H}_{S}^{2}\right)$. Because $0\le Co{s}_{HFL}^{\prime}\left({H}_{S}^{1},{H}_{S}^{2}\right)\le 1$, ${D}_{HFL}^{\prime}\left({H}_{S}^{1},{H}_{S}^{2}\right)\ge 0$ hold simultaneously, then we have ${D}_{HFL}^{\prime}\left({H}_{S}^{1},{H}_{S}^{2}\right)=0$, $Co{s}_{HFL}^{\prime}\left({H}_{S}^{1},{H}_{S}^{2}\right)=1$. When $Co{s}_{HFL}^{\prime}\left({H}_{S}^{1},{H}_{S}^{2}\right)=1$, we know that ${H}_{S}^{1}=k{H}_{S}^{2}$ and $k$ is a constant; while ${D}_{HFL}^{\prime}\left({H}_{S}^{1},{H}_{S}^{2}\right)=0$, we know that ${H}_{S}^{1}={H}_{S}^{2}$. That is to say, when ${S}_{HFL}^{\ast}\left({H}_{S}^{1},{H}_{S}^{2}\right)=1$, ${H}_{S}^{1}={H}_{S}^{2}$.Thus, ${S}_{HFL}^{\ast}\left({H}_{S}^{1},{H}_{S}^{2}\right)=1$ if and only if ${H}_{S}^{1}={H}_{S}^{2}$.
- (3)
- According to Remark 5, $Co{s}_{HFL}^{\prime}\left({H}_{S}^{1},{H}_{S}^{2}\right)=Co{s}_{HFL}^{\prime}\left({H}_{S}^{2},{H}_{S}^{1}\right)$ is obvious. From Theorem 2, it is already known that when ${D}_{HFL}^{\prime}\left({H}_{S}^{1},{H}_{S}^{2}\right)={D}_{HFL}^{\prime}\left({H}_{S}^{2},{H}_{S}^{1}\right)$, then ${S}_{HFL}^{\ast}\left({H}_{S}^{1},{H}_{S}^{2}\right)\text{}={S}_{HFL}^{\ast}\left({H}_{S}^{2},{H}_{S}^{1}\right)$ are proven. ☐

**Remark**

**6.**

**Theorem**

**4.**

- (1)
- $0\le {D}_{HFL}^{\ast}\left({H}_{S}^{1},{H}_{S}^{2}\right)\le 1$;
- (2)
- ${D}_{HFL}^{\ast}\left({H}_{S}^{1},{H}_{S}^{2}\right)=0$if and only if${H}_{S}^{1}={H}_{S}^{2}$;
- (3)
- ${D}_{HFL}^{\ast}\left({H}_{S}^{1},{H}_{S}^{2}\right)={D}_{HFL}^{\ast}\left({H}_{S}^{2},{H}_{S}^{1}\right)$.

**Proof.**

**Definition**

**11.**

**Theorem**

**5.**

**Proof.**

**Remark**

**7.**

**Remark**

**8.**

**Example**

**5.**

## 4. The TOPSIS Method with the Proposed Distance Measure ${\mathit{D}}_{\mathit{\omega}\mathit{H}\mathit{F}\mathit{L}}^{\mathbf{\ast}}$

**Step 1.**Normalize the hesitant fuzzy linguistic decision matrix $H$.

**Step 2.**For $i=1,2,\cdots ,\text{}m,j=1,2,\cdots ,\text{}n$, the hesitant fuzzy linguistic positive ideal solution (HFLPIS) ${H}^{+}=\{{H}_{S}^{1+},{H}_{S}^{2+},\cdots ,{H}_{S}^{n+}\}$ and hesitant fuzzy linguistic negative ideal solution (HFLNIS) ${\mathrm{H}}^{-}=\{{\mathrm{H}}_{\mathrm{S}}^{1-},{\mathrm{H}}_{\mathrm{S}}^{2-},\cdots ,{\mathrm{H}}_{\mathrm{S}}^{\mathrm{n}-}\}$ are given in the following:

**Step 3.**Use the distance measure to calculate the separation of each alternative between the HFLPIS ${H}^{+}=\{{H}_{S}^{1+},{H}_{S}^{2+},\cdots ,{H}_{S}^{n+}\}$ and HFLNIS ${H}^{-}=\{{H}_{S}^{1-},{H}_{S}^{2-},\cdots ,{H}_{S}^{n-}\}$, respectively.

**Step 4.**Calculate the closeness coefficient ${\mathsf{\Phi}}_{i}$ of each alternative ${H}_{i}\left(i=1,2,\cdots ,m\right)$:

**Step 5.**Rank the alternatives by decreasing order of the closeness coefficient ${\mathsf{\Phi}}_{i}$; the greater value ${\mathsf{\Phi}}_{i}$ is, the better alternative ${H}_{i}$ will be.

## 5. Numerical Example

#### 5.1. Background

**Step 1.**Normalize the hesitant fuzzy linguistic decision matrix.

**Step 2.**According to the score function in Theorem 1, we can calculate the HFLPIS ${\mathrm{H}}^{+}$ and the HFLNIS ${\mathrm{H}}^{-}$, which are given as follows:

**Step 3.**Calculate the distance measure ${D}_{\omega HFL}^{\ast}({H}_{S}^{ij},{H}^{+})$ and ${D}_{\omega HFL}^{\ast}({H}_{S}^{ij},{H}^{-})$ for different alternative ${H}_{i}\left(i=1,2,3,4,5\right)$ respectively, which are given in Table 5.

**Step 4.**Calculate the closeness coefficient ${\mathsf{\Phi}}_{i}$ of each alternative ${H}_{i}$; they are obtained in Table 6.

**Step 5.**Rank the alternatives ${H}_{i}$ and utilize ${\mathsf{\Phi}}_{i}\left(i=1,2,3,4,5\right)$.

#### 5.2. Comparison Analysis

- (1)
- The distance measure ${D}_{\omega HFL}^{\ast}$ is derived from the cosine function and the Euclidean distance measure; it considers the distance measure not only from the point of view of algebra, but also from the point of view of geometry. It shows a better performance when the subscripts of the linguistic term sets in the two HFLTS have the linear relationship.
- (2)
- The similarity measure ${S}_{\omega HFL}^{\ast}$ and distance measure ${D}_{\omega HFL}^{\ast}$ based on the linguistic scale function can express information better under different circumstances, and the decision makers can select the appropriate linguistic scale function $f$ on the basis of their preferences. It also can be applied more widely in the decision-making field than the existing distance measure and cosine similarity measure.
- (3)
- The proposed method focuses on the differences of each alternative, which can improve the effectiveness of solving MCDM problems.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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Viral Fever | Typhoid | Pneumonia | Stomach Problem | |
---|---|---|---|---|

Richard | {s_{5}} | {s_{5}} | {s_{4},s_{5}} | {s_{0}} |

Catherine | {s_{3}} | {s_{0}} | {s_{0}} | {s_{4},s_{5}} |

Nicle | {s_{6}} | {s_{4}} | {s_{5}} | {s_{0}} |

Kevin | {s_{4}} | {s_{2},s_{3}} | {s_{5}} | {s_{0}} |

Viral Fever | Typhoid | Pneumonia | Stomach Problem | |
---|---|---|---|---|

Richard | {s_{4},s_{5},s_{6}} | {s_{3},s_{4},s_{5}} | {s_{4},s_{5},s_{6}} | {s_{0}} |

Catherine | {s_{5},s_{6}} | {s_{1},s_{2},s_{3}} | {s_{4},s_{5},s_{6}} | {s_{0},s_{1}} |

Nicle | {s_{3},s_{4}} | {s_{2},s_{3}} | {s_{5},s_{6}} | {s_{0}} |

Kevin | {s_{3}} | {s_{0}} | {s_{0}} | {s_{4},s_{5},s_{6}} |

Viral Fever | Typhoid | Pneumonia | Stomach Problem | |
---|---|---|---|---|

Richard | 0.9574 | 0.9702 | 0.8973 | 0.8606 |

Catherine | 0.7865 | 0.7856 | 0.8323 | 0.9954 |

Nicle | 0.9558 | 0.9517 | 0.8846 | 0.7741 |

Kevin | 0.9279 | 0.8728 | 0.9814 | 0.8370 |

${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ | ${\mathit{C}}_{3}$ | ${\mathit{C}}_{4}$ | |
---|---|---|---|---|

${H}_{1}$ | {s_{2},s_{3},s_{4}} | {s_{3},s_{6}} | {s4,s6} | {s_{0},s_{1},s_{2}} |

${H}_{2}$ | {s_{3},s_{4}} | {s_{4},s_{6}} | {s_{0},s_{1}} | {s_{1},s_{4}} |

${H}_{3}$ | {s_{0},s_{1}} | {s_{4}} | {s_{0},s_{1},s_{3}} | {s_{2}} |

${H}_{4}$ | {s_{5}} | {s_{1},s_{3}} | {s_{4},s_{6}} | {s_{0},s_{1},s_{4}} |

${H}_{5}$ | {s_{4},s_{5}} | {s_{2},s_{3}} | {s_{1},s_{3},s_{4}} | {s_{0},s_{2}} |

${\mathit{D}}_{\mathit{i}}^{+}$ | ${\mathit{D}}_{\mathit{i}}^{-}$ | |
---|---|---|

${H}_{1}$ | 0.1524 | 0.2907 |

${H}_{2}$ | 0.2395 | 0.3169 |

${H}_{3}$ | 0.4201 | 0.1694 |

${H}_{4}$ | 0.1753 | 0.4624 |

${H}_{5}$ | 0.1884 | 0.3740 |

${\mathit{H}}_{1}$ | ${\mathit{H}}_{2}$ | ${\mathit{H}}_{3}$ | ${\mathit{H}}_{4}$ | ${\mathit{H}}_{5}$ | |
---|---|---|---|---|---|

${\mathsf{\Phi}}_{i}$ | 0.6561 | 0.5695 | 0.2873 | 0.7251 | 0.6650 |

Ranking | |
---|---|

$f={f}_{1}\left({s}_{i}\right)$ | ${H}_{4}\succ {H}_{5}\succ {H}_{1}\succ {H}_{2}\succ {H}_{3}$ |

$f={f}_{2}\left({s}_{i}\right)$ | ${H}_{4}\succ {H}_{1}\succ {H}_{5}\succ {H}_{2}\succ {H}_{3}$ |

$f={f}_{3}\left({s}_{i}\right)$ | ${H}_{4}\succ {H}_{5}\succ {H}_{1}\succ {H}_{2}\succ {H}_{3}$ |

Ranking | |
---|---|

Approach from Liao et al. [9] | ${H}_{2}\succ {H}_{1}\succ {H}_{4}\succ {H}_{5}\succ {H}_{3}$ |

Approach from Liao et al. [31] | ${H}_{4}\succ {H}_{5}\succ {H}_{1}\succ {H}_{2}\succ {H}_{3}$ |

Approach from Wang et al. [38] | ${H}_{4}\succ {H}_{1}\succ {H}_{2}\succ {H}_{5}\succ {H}_{3}$ |

Approach from Zhang et al. [41] | ${H}_{4}\succ {H}_{1}\succ {H}_{2}\succ {H}_{5}\succ {H}_{3}$ |

Proposed approach based on ${D}_{\omega HFL}^{\ast}$ | ${H}_{4}\succ {H}_{5}\succ {H}_{1}\succ {H}_{2}\succ {H}_{3}$ |

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## Share and Cite

**MDPI and ACS Style**

Liu, D.; Liu, Y.; Chen, X.
The New Similarity Measure and Distance Measure of a Hesitant Fuzzy Linguistic Term Set Based on a Linguistic Scale Function. *Symmetry* **2018**, *10*, 367.
https://doi.org/10.3390/sym10090367

**AMA Style**

Liu D, Liu Y, Chen X.
The New Similarity Measure and Distance Measure of a Hesitant Fuzzy Linguistic Term Set Based on a Linguistic Scale Function. *Symmetry*. 2018; 10(9):367.
https://doi.org/10.3390/sym10090367

**Chicago/Turabian Style**

Liu, Donghai, Yuanyuan Liu, and Xiaohong Chen.
2018. "The New Similarity Measure and Distance Measure of a Hesitant Fuzzy Linguistic Term Set Based on a Linguistic Scale Function" *Symmetry* 10, no. 9: 367.
https://doi.org/10.3390/sym10090367